Hilbert spaces

Abstract

The goal of this chapter is to give an elementary but comprehensive introduction to the theory of Hilbert spaces, where we wish to highlight the many-faceted interplay of their geometric and analytic properties. This culminates in the theorem of Riesz–Fréchet, which describes continuous linear forms in a Hilbert space. Important for us will be that this theorem can be interpreted as an existence and uniqueness result. A generalization known as the Lax–Milgram theorem, which we will give in Section 4.5, is at the heart of the solution theory of elliptic equations which we will present in Chapters 5, 6 and 7. For the reader who is principally interested in these equations, the part of the current chapter up to and including Section 4.5 provides sufficient background.